Semilocal convergence analysis for the modified Newton-HSS method under the Hölder condition

We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] in combination with the weaker center 2-piece L1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal.22 (2002) 359–390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math.8 (2008) 197–226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant α-theory, IMA J. Numer. Anal.23 (2003) 395–419].

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Semilocal convergence analysis of a fifth-order method using recurrence relations in Banach space under weak conditions

Applicationes Mathematicae ◽

10.4064/am2318-1-2017 ◽

2018 ◽

Vol 45 (2) ◽

pp. 223-231

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Recurrence Relations ◽

Semilocal Convergence ◽

Order Method ◽

Fifth Order

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A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2004.04.008 ◽

2004 ◽

Vol 298 (2) ◽

pp. 374-397 ◽

Cited By ~ 174

Author(s):

Ioannis K. Argyros

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Semilocal Convergence

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Semilocal convergence analysis on the modifications for Chebyshev–Halley methods under generalized condition

Applied Mathematics and Computation ◽

10.1016/j.amc.2016.01.035 ◽

2016 ◽

Vol 281 ◽

pp. 243-251 ◽

Cited By ~ 1

Author(s):

Xiuhua Wang ◽

Jisheng Kou

Keyword(s):

Convergence Analysis ◽

Semilocal Convergence

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Semilocal Convergence Theorem for the Inverse-Free Jarratt Method under New Hölder Conditions

The Scientific World JOURNAL ◽

10.1155/2015/346571 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

Yueqing Zhao ◽

Rongfei Lin ◽

Zdenek Šmarda ◽

Yasir Khan ◽

Jinbiao Chen ◽

...

Keyword(s):

Banach Space ◽

Error Estimate ◽

Convergence Analysis ◽

Operator Equation ◽

Convergence Theorem ◽

Nonlinear Operator ◽

Semilocal Convergence ◽

Nonlinear Operator Equation ◽

Jarratt Method ◽

Semilocal Convergence Theorem

Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem.

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Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

International Journal of Computational Methods ◽

10.1142/s0219876218500482 ◽

2018 ◽

Vol 15 (06) ◽

pp. 1850048

Author(s):

Sukhjit Singh ◽

Dharmendra Kumar Gupta ◽

Randhir Singh ◽

Mehakpreet Singh ◽

Eulalia Martinez

Keyword(s):

Convergence Analysis ◽

Banach Spaces ◽

Semilocal Convergence ◽

Fréchet Derivative ◽

Frechet Derivative ◽

First Order ◽

Weaker Conditions ◽

Fifth Order ◽

Derivatives Of ◽

Text Condition

The convergence analysis both local under weaker Argyros-type conditions and semilocal under [Formula: see text]-condition is established using first order Fréchet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Hölder conditions are particular cases of the [Formula: see text]-condition. Examples can be constructed for which the Lipchitz and Hölder conditions fail but the [Formula: see text]-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.

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EXTENDING THE APPLICABILITY OF NEWTON'S METHOD ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500411 ◽

2013 ◽

Vol 06 (03) ◽

pp. 1350041

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Error Analysis ◽

Convergence Analysis ◽

Lipschitz Condition ◽

Riemannian Manifolds ◽

Newton’S Method ◽

Newton's Method ◽

Semilocal Convergence ◽

Inclusion Problem ◽

Convergence Conditions ◽

Precise Information

We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math.13(2B) (2009) 633–656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton–Kantorovich theorem.

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